The Airborne Maneuvering Network (AMN) is a novel network architecture that enables flexible deployment across wide areas and provides real-time cross-domain transmission services. Under concurrent and diversified service demands, AMN operating in isolation faces significant challenges in guaranteeing end-to-end transmission reliability. This has prompted the deep integration of AMN with terrestrial and satellite networks to form heterogeneous networks, which has become a crucial trend in improving the continuity and reliability of AMN services. However, network heterogeneity, dynamic resource distribution, and the absence of a unified reliable transmission mechanism impose severe challenges on multi-domain cooperative scheduling and differentiated-service adaptation. This paper serves as a reference for global scholars engaged in thorough research on heterogeneous integrated AMN. It outlines the fundamental characteristics of AMN, reviews recent advances and challenges in communication-sensing-computation coordination, unified control adaptation, and service reliability assurance, and discusses design concepts and future evolution paths for heterogeneous integrated AMN architectures.